direct sum of vector bundles in nLab

Context

Bundles

Linear algebra

• Idea
• Definition
• Examples
• Properties
• Whitney summands of trivial vector bundles
• Characteristic classes of Whitney sums
• Related concepts
• References

Definition

(direct sum of topological vector bundles via total spaces)

Let

1. XX be a topological space,

2. E 1→π 1XE_1 \overset{\pi_1}{\to} X and E 2→π 2XE_2 \overset{\pi_2}{\to} X two topological vector bundles over XX.

Then the direct sum of vector bundles E 1⊕ XE 2→EE_1 \oplus_X E_2 \to E is the topological vector bundle whose total space is the topological subspace

E 1⊕ XE 2≔{(v 1,v 2)∈E 1×E 2|π 1(v 1)=π 2(v 2)}⊂E 1×E 2 E_1 \oplus_X E_2 \;\coloneqq\; \left\{ (v_1, v_2) \in E_1 \times E_2 \,\vert\, \pi_1(v_1) = \pi_2(v_2) \right\} \;\subset\; E_1 \times E_2

of the product topological space of the two total spaces, and whose projection map is

E 1⊕ XE 2 ⟶AAπAA X (v 1,v 2) ↦AAA π 1(v 1)=π 2(v 2). \array{ E_1 \oplus_X E_2 &\overset{\phantom{AA}\pi\phantom{AA}}{\longrightarrow}& X \\ (v_1,v_2) &\overset{\phantom{AAA}}{\mapsto}& \pi_1(v_1) = \pi_2(v_2) } \,.

For x∈Xx \in X the vector space structure on the fibers

(E 1⊕E 2) x≃(E 1) x⊕(E 2) x (E_1 \oplus E_2)_x \simeq (E_1)_x \oplus (E_2)_x

is the one on the direct sum of vector spaces.

Definition

(direct sum of topological vector bundles via transition functions)

Let XX be a topological space, and let E 1→XE_1 \to X and E 2→XE_2 \to X be two topological vector bundles over XX.

Let {U i⊂X} i∈I\{U_i \subset X\}_{i \in I} be an open cover with respect to which both vector bundles locally trivialize (this always exists: pick a local trivialization of either bundle and form the joint refinement of the respective open covers by intersection of their patches). Let

{(g 1) ij:U i∩U j→GL(n 1)}AAAandAAA{(g 2) ij:U i∩U j⟶GL(n 2)} \left\{ (g_1)_{i j} \colon U_i \cap U_j \to GL(n_1) \right\} \phantom{AAA} \text{and} \phantom{AAA} \left\{ (g_2)_{i j} \colon U_i \cap U_j \longrightarrow GL(n_2) \right\}

be the transition functions of these two bundles with respect to this cover.

For i,j∈Ii, j \in I write

(g 1) ij⊕(g 2) ij : U i∩U j ⟶ GL(n 1+n 2) x ↦AAA ((g 1) ij(x) 0 0 (g 2) ij(x)) \array{ (g_1)_{i j} \oplus (g_2)_{i j} &\colon& U_i \cap U_j &\longrightarrow& GL(n_1 + n_2) \\ && x &\overset{\phantom{AAA}}{\mapsto}& \left( \array{ (g_1)_{i j}(x) & 0 \\ 0 & (g_2)_{i j}(x) } \right) }

be the pointwise direct sum of these transition functions

Then the direct sum bundle E 1⊕E 2E_1 \oplus E_2 is the one glued from this direct sum of the transition functions (by this construction):

E 1⊕E 2≔((⊔iU i)×(ℝ n 1+n 2))/({(g 1) ij⊕(g 2) ij} i,j∈I). E_1 \oplus E_2 \;\coloneqq\; \left( \left( \underset{i}{\sqcup} U_i \right) \times \left( \mathbb{R}^{n_1 + n_2} \right) \right)/ \left( \left\{ (g_1)_{i j} \oplus (g_2)_{i j} \right\}_{i,j \in I} \right) \,.

Example

Let XX and YY be topological spaces, and write X⊔YX \sqcup Y for their disjoint union space.

Then every topological vector bundle on X⊔YX \sqcup Y is the direct sum of a vector bundle that has rank zero on YY and one that has rank zero on XX.

More explicitiy: let

i X:Vect(X)⟶Vect(X⊔Y) i_X \colon Vect(X) \longrightarrow Vect(X \sqcup Y)

and

i Y:Vect(Y)⟶Vect(X×Y) i_Y \colon Vect(Y) \longrightarrow Vect(X \times Y)

be the operations of extending a vector bundle on the other connected component by a rank-0 vector bundle, then

Vect(X)×Vect(Y)⟶≃i X⊕ (X⊔Y)i YVect(X⊔Y) Vect(X) \times Vect(Y) \underoverset{\simeq}{ i_X \oplus_{(X \sqcup Y)} i_Y }{\longrightarrow} Vect(X \sqcup Y)

is an isomorphism of isomorphism classes of vector bundles (and an equivalence of categories of categories of vector bundles before passing to isomorphism classes).

Proof

Since XX is assumed to be paracompact Hausdorff, there exists a inner product on vector bundles

⟨−,−⟩:E⊕ XE⟶X×ℝ \langle -,-\rangle \;\colon\; E \oplus_X E \longrightarrow X \times \mathbb{R}

(by this prop.). This defines at each x∈Xx \in X the orthogonal complement (E′ x) ⊥⊂E x(E'_x)^\perp \subset E_x of E′ x↪EE'_x \hookrightarrow E. The subspace of these orthogonal complements is readily checked to be a topological vector bundle (E′) ⊥→X(E')^\perp \to X. Hence by construction we have

E≃E′⊕ X(E′) ⊥. E \;\simeq\; E' \oplus_X (E')^\perp \,.

Proof

Let {U i⊂X} i∈I\{U_i \subset X\}_{i \in I} be an open cover of XX over which E→XE \to X has a local trivialization

{ϕ i:U i×ℝ n⟶≃E| U i} i∈I. \left\{ \phi_i \;\colon\; U_i \times \mathbb{R}^n \overset{\simeq}{\longrightarrow} E\vert_{U_i} \right\}_{i \in I} \,.

By compactness of XX, there is a finite sub-cover, hence a finite set J⊂IJ \subset I such tat

{U i⊂X} i∈J⊂I \{U_i \subset X\}_{i \in J \subset I}

is still an open cover over which EE trivializes.

Since paracompact Hausdorff spaces equivalently admit subordinate partitions of unity there exists a partition of unity

{f i:X→[0,1]} i∈J \left\{ f_i \;\colon\; X \to [0,1] \right\}_{i \in J}

with support supp(f i)⊂U isupp(f_i) \subset U_i. Hence the functions

E| U i ⟶AAAA U i×ℝ n v ↦AAA f i(x)⋅ϕ i −1(v) \array{ E\vert_{U_i} &\overset{\phantom{AAAA}}{\longrightarrow}& U_i \times \mathbb{R}^n \\ v &\overset{\phantom{AAA}}{\mapsto}& f_i(x) \cdot \phi_i^{-1}(v) }

extend by 0 to vector bundle homomorphism of the form

f i⋅ϕ i −1:E⟶X×ℝ n. f_i \cdot \phi^{-1}_i \;\colon\; E \longrightarrow X \times \mathbb{R}^n \,.

The finite pointwise direct sum of these yields a vector bundle homomorphism of the form

⊕i∈Jf i⋅ϕ i:E⟶X×(⊕i∈Jℝ n)≃X×ℝ n|J|˙. \underset{i \in J}{\oplus} f_i \cdot \phi_i \;\colon\; E \longrightarrow X \times \left( \underset{i \in J}{\oplus} \mathbb{R}^n \right) \simeq X \times \mathbb{R}^{n \dot {\vert J\vert}} \,.

Observe that, as opposed to the single f i⋅ϕ i −1f_i \cdot \phi^{-1}_i, this is a fiber-wise injective, because at each point at least one of the f if_i is non-vanishing. Hence this is an injection of EE into a trivial vector bundle.

With this the statement follows by prop. .